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Again we return, in our imaginations, to the world in which light flies one hundred miles in an hour and now we have a train moving eastward (the positive x-direction) at a speed V on a straight track. On a flat car we have mounted a pair of identical wheels, each of which has all of its mass m0 concentrated in its rim, a distance R from the axle, which we have oriented parallel to the track. We have so geared the wheels together that the electric motor that drives them applies to them equal torques in opposite senses: the wheels thus gain no net angular momentum, regardless of how fast they spin. We also provide a battery to supply electricity to the motor.
At some time an experimenter aboard the train turns on the electric motor and makes the wheels spin faster, making their rims move faster by an increment of speed dv. This act makes the wheels each gain an increment of mass
In the frame occupied by a trackside observer that increment corresponds to
In gaining that increment of mass each wheel also gains an increment of linear momentum dm'V in the x-direction and thus manifests a force in the x-direction. But where shall we find the necessary equal and oppositely directed force required to fulfill Newton's third law of motion? We won't find it in any deceleration of the train because the train does not decelerate when the wheels spin up: indeed, the force attending the wheels' increase of mass does not exist in the train's inertial frame. Thus, we must infer that whatever phenomenon produces the Newtonian reaction to that phantom force also generates no force in the train frame.
We thus assert that some part of our apparatus suffers a decrease in mass when the wheels spin up and that the decrease of mass equals the increase of mass imposed on the wheels. That decrease makes the phantom force add up to a net zero over the whole apparatus. But what necessarily loses mass when the wheels spin up?
We know that in order to spin the wheels up the motor that drives them must necessarily draw electrical energy from the battery and convert it into kinetic energy in the spinning wheels. That fact points our attention to the only feature of our system whose change is necessary to the spinning up of the wheels, so we infer that the energy stored in the battery ponders mass and, thus, that the movement of energy from the battery to the wheels via the motor comprises a mere rearrangement of the system's mass. But now we must ask How much mass does a given amount of energy ponder? We can answer that question by determining the amount of work that the motor must do in spinning up one of the wheels.
Work comes from exerting a force over a distance (dW=Fds). We apply that fact to one of the wheels by noting that the torque exerted by the motor acts on the wheel as if we had uniformly applied a force around the circle representing the wheel's rim. Mathematically, then, we can represent the torque increasing the rotary motion of the wheel as if we had manifested it as a simple force accelerating a body in a straight line. Thus we have, as calculated by the trackside observer,
If we identify that work done with the energy that the wheel gains (dW = dE) as it gains mass dm' and if we then compare Equation 2 with Equation 3, we see that
the differential form of the most famous equation ever written.
We thus correlate the amount of mass that the wheels gain with the energy that the motor must put into the wheels to produce that gain. But the conservation law tells us that the battery must lose exactly as much energy as the wheels gain and the necessity that the phantom force acting on the apparatus add up to a net zero necessitates that the battery lose as much mass as the wheels gain, so Equation 2 applies to the battery as well as to the wheels. Those facts tell us that we can now generalize our result and state that energy, in whatever form we manifest it, confers upon its container a mass in accordance with Equation 4.
Conversely, we can assert that every bit of mass in a body corresponds to a certain amount of energy, however unavailable that energy may be for our purposes. For a body of rest mass m0 moving with speed v we can write
Square that equation and multiply it by 1-v2/c2 and we obtain
Substituting E=mc2 into the second term on the left side of the equality sign and recognizing mv=p as representing the body's linear momentum gives us
the form of Equation 4 that physicists use as an accounting tool when they study elementary-particle interactions.
Contrast that way of deducing the famous mass-energy equivalence with the logical path that Einstein followed. In his 1907 paper on the subject Einstein imagined a body that emitted identical beams of light in opposite directions. He then feigned observing that body from an inertial frame in which the body moved. As an observer in that second frame he calculated the Doppler shift of the waves in the light beams and from that the power flowing in the beams. In that frame one beam, red-shifted, would carry less power than the other, blue-shifted beam, so it would exert less force on the body as it decoupled from the body's matter. The difference between those forces could not accelerate the body, because, Einstein knew, the body did not accelerate in its rest frame, where the beams carried the same amount of power and thus exerted equal forces that canceled each other.
Inferring that the force exerted upon the body by the two opposing beams could only conform to the force produced by a change in a body's mass (the rocket-thrust part of the equation relating force to the rate at which a body's linear momentum changes), Einstein then inferred that the mass of the body must have been changing at a rate that he could calculate by dividing the force that the two beams exerted upon the body by the speed of the body in his second inertial frame. Correlating that mass with the rate at which the body was putting energy into the beams then gave him his famous equation.
In deducing the laws of relativistic energy I have led you down a logical path different from the one that Einstein followed in deducing those rules originally, because I want to continue to feign at this stage that I do not know the true nature of light. I want to pretend that I know only that light exists and that it carries energy because I want to see whether I can deduce its nature from the constraints that the fundamental laws of nature impose upon it.
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